Forwards & Futures
Regular Forward
You agree now to buy an asset at time t and the money is exchanged at time t. This equals what we expect the stock to be worth at time t.
- \(F=S_0 e^{(r-\delta)t}\)
- Continuous Dividends
- Compound by r, discount by \(\delta\) b/c you aren’t receiving the dividends
- Remember to always compound by r, even if you are given the expected return of the stock (\(\alpha\))
- \(F=[S_0 - \sum (Div_i)(e^{-rt_i})]e^{rt}\)
- Discrete Dividends
- Discount each dividend to the PV so you get them all at the same time, then compound everything to time t. We use the rfr for all discounting and compounding here.
Prepaid Forward
You pay upfront to receive the asset at time t. Asset holder still receives dividends during this time.
- \(F^P = S_0e^{-\delta t}\)
- Continuous dividends
- Just discount the stockprice by the dividends b/c you don’t get these
- \(F^P = S_0 - \sum (Div_i)(e^{-rt_i})\)
- Discrete dividends are discounted by the continuous rfr rate
Symbols
- \(F\): Forward price you pay at time t
- \(F^P\): Prepaid forward price you pay at time 0
- \(S_0\) or just \(S\): Stock price at t=0
- \(r\): Risk free interest rate
- \(\delta\): continuously compounded dividend rate
- \(t\): time until expiration
Relationship between regular and prepaid forwards
- \(F = F^Pe^{rt}\)
- Regular forwards equations are just the prepaid forward equations compounded by \(e^{rt}\)
Currency Forwards
- Think of currency forwards just like asset forwards. But that you are receiving a foreign currency as the asset. The symbols and language is different but you can still use the above formulas.
- \(F = x_0e^{(r-r_f)t}\)
- \(F^P = x_0e^{-r_f(t)}\)
- \(x_0\) = Exchange Rate, how much you have to pay in base currency to receive the foreign currency. (\(S_0\))
- \(r\) = rfr for the base currency you currently have
- \(r_f\) = rfr for the foreign currency you are buying (\(\delta\))
Futures
- Futures use the same formulas as forwards
- Futures are traded on exchanges while forwards are traded over the counter
- Futures are cash settled, and deal with margins
- Some language
- Notional Value = Index * Size
- Margin = %age of notional value that you have to keep in your margin account (in case you default)
- Maintenance Margin = %age of your original margin balance that is the minimum allowed amount. You will get a marign call if you have less money than the maintenance margin and have to deposit more money.
- Ex: You enter into a futures contract to long the S&P 500. The size is $200, the index is 700, margin is 25%, maintenance margin is 80%, and risk free rate is .05.
- Balances at t=0:
- Notional Value = Index * Size = \(\$700(200) = \$140,000\)
- Initial balance in the margin account = (Notional Value)(margin %) = \(\$140,000(.25) = \$35,000\)
- Minimum balance before a margin call = \(\$35,000(.8) = \$28,000\)
- 1 week passes. Index price rises to 730.
- Margin account balance grows by rfr for 1 week: \(\$35,000e^{.05(\frac{1}{52})}=35033.67\)
- Index price rose by \(730-700=30\), so margin balance increases by \(30*200=6,000\)
- Margin balance at t = 1 week:
- \(=\$35,000e^{.05(\frac{1}{52})} + 6,000=41033.67\)
- Another week passes. t = 2 weeks. Index price falls to $650.
- Margin grows by rfr, and margin balance decreases by \(730-650=\$80\) times the size of 200
- Margin Balance at t = 2 weeks \(=\$4.103367\times 10^{4}e^{(\frac{.05}{52})} - (80)(200)=25073.14\)
- So we would get a margin call to add more money to the margin account since the value dropped below the maintenance margin of $28,000
- Balances at t=0:
- If you are short the index, then when the index rises this is a bad thing, and your margin account should decrease.
- For ex, take the week 1 scenario above when the index rises to 730. Instead of growing the margin balance by \(30*200=6,000\), we would decrease the margin balance by $6k.
Definitions
- Forward Premium = \(\frac{F}{S_0}\)
- Annualized Forward Premium = \(\frac{1}{t}Log(\frac{F}{S_0})=r-\delta\)
- (Expressed as a percentage)
- Risk Premium = \(\alpha - r\)
- Rate of Appreciation = \(\alpha - \delta\)
- AKA: Continuously compounded expected rate of appreciation
- Cost of Carry = \(r-\delta\)
Options Basics
Overview
- Main types
- European = can only be exercised at expiration, easier math so assume this
- American = can be exercised at any point prior to expiration, must use binomial trees for pricing this
- Exotic Types
- Asian, Barrier, Compound, Gap, Exchange
- Payoffs
- Long Call = max{0, Spot - Strike}
- Buying the option to buy the stock at the strike price
- Hoping stock goes up, so you can buy at low strike, and sell at high spot
- Long Put = max{0, Strike - Spot}
- Buying the option to sell the stock at the strike price
- Hoping stock goes down, so you can buy the stock at the low spot, and sell at a high strike
- Short Call = min{0, Strike - Spot}
- Selling someone the option to buy the stock from you at the strike price
- Hoping the stock will go down so the option you sold is never exercised w/ 0 payoff. So you just profit the premium.
- Short Put = min{0, Spot - Strike}
- Selling someone the option to sell you the stock at the strike price
- Hoping the stock goes up, so the spot > strike, so they won’t sell it to you at the lower price. So you just profit the premium.
- Long Call = max{0, Spot - Strike}
- Profits
- = payoff + FV of the premium (compounded by the rfr)
- Language
- Spot price: actual price of the stock at expiration
- Strike: agreed upon price that will be exchanged for the asset
General strategy for dealing with solo call/put payoffs
- Look at the left and right tails
- If the LEFT side goes up/down => Put
- If the RIGHT side goes up/down => Call
- If it goes UP => Long
- If it goes DOWN => Put
Options Profit Plots
Option + Asset Combinations
Floor
Cap
Covered Call
Covered Put
Options Spreads
- Spreads mean it’s using all calls or all puts. But this definition is fudged a bit.
Bull Spread
- 2 Calls OR 2 Puts.
- Buy LOW & Sell HIGH
Bear Spread
- 2 Calls OR 2 Puts
Buy HIGH & Sell LOW
2 Calls OR 2 Puts
Box Spread
- Rare to see on IFM
- Box spreads are a 0 profit spread, so it’s the same as lending/borrowing money
- A 4 option strategy consisting of a Bull Spread + Bear Spread
- 2 ways to build, and this changes whether it’s equivalent to borrowing or lending money
- Bull Spread w/ Calls + Bear Spread w/ Puts => Lending money at rfr (buying bonds)
- Bull Spread w/ Puts + Bear Spread w/ Calls => Borrowing money at rfr (shorting bonds)
- All using only 2 strikes (\(K_1\) & \(K_2\))
Ratio Spread
- Combination of buying/selling m options at one strike price, and n options at a different strike price.
Collars
Purchased Collar
- BUY Put low, WRITE Call High.
- So it’s flat in the middle
- Collar Width: Distance between the Put and the Call
- Weird that a Purchased collar looks like a bearish position
Written Collar
- WRITE Put Low, LONG Call High
Zero Cost Collar
- A Collar Spread (either purchased or written) that you break even (0 profit) in the middle
Long Straddle
- Buy a call and a put at the same strike price
- Betting on high volatility
Written Straddle
- Writting both a call and a put at the same strike price
- Betting on low volatility
Strangle
- Long both a call and a put at different strike prices (it doesn’t matter which one is low/high)
Written Strangle
- Write both a call and a put at different strike prices
Butterfly Spread
Combination of 4 option positions at 3 different strike prices (\(K_1 < K_2 < K_3\))
For a purchased Butterfly Spread, there are several ways to construct.
- Main way to construct (using all calls or all puts):
- All Calls:
- Buy \(C(K_1)\)
- Sell 2 \(C(K_2)\)
- Buy \(C(K_3)\)
- All Puts:
- Buy \(P(K_1)\)
- Sell 2 \(P(K_2)\)
- Buy \(P(K_3)\)
- Another way to think of this is a combination of a Bull & Bear spread
- All Calls:
- Using a combination of calls and puts:
- Buy \(P(K_1)\)
- Sell \(P(K_2)\)
- Sell \(C(K_2)\)
- Buy \(C(K_3)\)
- Another way to think of this is to write a Straddle at \(K_2\) + Buy a Strangle w/ strikes \(K_1\) & \(K_3\)
- When constructed this way it is called an “Iron Butterfly”
Each of these positions create this identical plot:
- Written Butterfly
- Switch long/short positions from above
Asymmetric Butterfly
- Still work w/ 3 strike prices \(K_1 < K_2 < K_3\)
- But this time \(K_2\) isn’t exactly in the middle
Coaching Actuaries Way:
- Long (\(K_3-K_2\)) options at \(K_1\)
- Short (\(K_3-K_1\)) options at \(K_2\)
- Long (\(K_2-K_1\)) options at \(K_3\)
- How many you long/short at a certain strike price, is the difference between the other 2 strike prices
- Much simpler than the method below that comes from the textbook:
Textbook way:
To construct an asymetric butterfly w/ all calls:
You have some \(\lambda\) where:
- \(\lambda = \frac{K_3 - K_2}{K_3 - K_1}\)
- Solving for \(K_2 = \lambda K_1 + (1- \lambda)K_3\)
- So for every \(K_2\) Call you write, you must:
- Buy \(\lambda K_1\) Calls
- Buy \((1-\lambda) K_3\) Calls
Ex: Suppose you wish to create an asymetric butterfly with strikes 100, 110, and 115. You wish to buy/sell 12 options total. How many of each option do you buy/sell?
- \(\lambda = \frac{K_3 - K_2}{K_3 - K_1} = \frac{115 - 110}{115 - 100} = \frac{1}{3}\)
So our ratios are as follows:
- 1 \(K_2\) Call sold
- 1/3 \(K_1\) calls bought
- 2/3 \(K_3\) calls bought
Want 12 options total:
- \(1x + \frac{1}{3}x + \frac{2}{3}x = 12\)
- \(x = 6\)
Multiply the above ratios by 6 each to see how many of each to buy
Iron Condor
- Like a butterfly, but with space between the options at the point. So it has 4 strikes \(K_1 < K_2 < K_3 < K_4\)
- Not even sure if this is tested on IFM but it’s easy
Synthetic Forward
- Buying a call and Selling a put at the same strike price w/ the same expiration mimics a forward contract.
Arbitrage
Random things to know:
- American options should be worth more than European options, b/c they have the added flexibility to exercise at anytime.
- Never rational to exercise an American Call early on a non dividend paying stock
- B/c the only benefit to exercising early is to receive the high dividends, but you could be earning risk free interest in the mean time if there are no dividends.
- American Call Prem = European Call Prem, when there are no dividends
- When American options make sense to exercise early:
- American Calls make sense to exercise early when: PV(Dividends) > PV(Interest on Strike) + PV(Insurance)
- American Puts make sense to exercise early when: PV(Interest on Strike) > PV(Dividends) + PV(Insurance)
- PV(Insurance) means the premium
- You exercise early when you can recive more on the dividends or interest basically
3 cases when arbitrage exists
| Proposition | Calls | Puts |
|---|---|---|
| 1 | \(C(K_1) \geq C(K_2)\) | \(P(K_1) \leq P(K_2)\) |
| 2 | \(C_{Amer}(K_1)-C_{Amer}(K_2) \leq K_2 -K_1\) \(C_{Eur}(K_1)-C_{Eur}(K_2) \leq (K_2-K_1)e^{-rT}\) |
\(P_{Amer}(K_2)-P_{Amer}(K_1) \leq K_2 -K_1\) \(P_{Eur}(K_2)-P_{Eur}(K_1) \leq (K_2-K_1)e^{-rT}\) |
| 3 | \(\frac{C(K_1)-C(K_2)}{K_2-K_1} \geq \frac{C(K_2)-C(K_3)}{K_3-K_2}\) \(C(K_2)\leq \lambda C(K_1) + (1-\lambda)C(K_3)\) |
\(\frac{P(K_2)-P(K_1)}{K_2-K_1} \leq \frac{P(K_3)-P(K_2)}{K_3-K_2}\) \(P(K_2) \leq \lambda P(K_1) + (1-\lambda)P(K_3)\) |
In Words:
- Self explanatory, premiums rise when more in the money
- Premiums move slower than strikes. If you increase the strike by $5, the premium will change by less than $5.
- \(\Delta\)Premium < \(\Delta\)Strike
- A certain convexity exists. When more in the money, the ratio of difference of premiums to the difference of strikes will be greater (closer to 1). Remember than when you move way deep into the money, the profit becomes almost guaranteed, so the change in premium will almost equal the change in strike.
- Pearson textbook formula for this theorem
- \(V(K_2) \leq \lambda V(K_1)+(1-\lambda)V(K_3)\)
- \(\lambda = \frac{K_3-K_2}{K_3-K_1}\)
- To exploit this, we short what’s more expensive, and long what’s cheaper.
- \(V(K_2) \leq \lambda V(K_1)+(1-\lambda)V(K_3)\)
- Pearson textbook formula for this theorem
- To exploit the arbitrage in any of the above formulas:
- Move everything to the greater than side of the equation (formulas > 0).
- \(+\) values => Sell/Borrow (getting money)
- \(-\) values -> Buy/Lend (spending money)
- Ex: \(P(65)-P(60)-5>0\)
- We Sell P(65), Buy P(60), and Lend $5 (buy bond)
Option Pricing
Basic Profit Functions
- Profit Long Call = \(Max[0, S_T - K] - FV(Prem)\)
- Profit Short Call = \(-Max[0, S_T - K] + FV(Prem)\)
- Profit Long Put = \(Max[0, K - S_T] - FV(Prem)\)
Profit Short Put = \(-Max[0, K-S_T]+FV(Prem)\)
- Short & Long of each are just multiplied by -1
- Calls are \(S_T-K\), and Puts are \(K-S_T\)
Put Call Parity
This parity must exist between the Call and Put premium for the same asset with the same strikes and expiration dates, or else arbitrage exists.
For Stocks
- \(C - P = Se^{-\delta t} - Ke^{-rt}\)
- \(C - P + Ke^{-rt} = Se^{-\delta t}\)
- Can rearrange so it looks like this. The left side of the equation is a synthetic prepaid forward, the right side of the equation is the actual prepaid forward.
For currency
- \(C-P = x_0e^{-r_f t}-Ke^{-rt}\)
- Symbols
- C = Call Premium
- P = Put Premium
- S = \(S_0\) Stock price at t=0
- \(x_0\) = Foreign currency price at t=0
- \(\delta\) = dividend yield of stock
- \(r_f\) = rfr of foreign currency (\(\delta\))
- Just use 1st equation and treat foreign currency like the stock
Binomial Trees
- AKA “The Usual Method in McDonald”, “The Standard Method”, “A Tree based on forward prices”
1 Period Binomial Tree
Assume stock can either rise from \(S_0\) to \(S_u\), or go down to \(S_d\) at time t.
- Option Premium = \(\Delta S + B\)
- \(\Delta = e^{-\delta t} \frac{V_u - V_d}{S(u - d)}\)
- \(B=e^{-rt} \frac{u V_d - d V_u}{u-d}\)
- \(V_u/V_d\) = Option payoff if stock rises/drops. For call or put
- \(S\) = \(S_0\) = Stock price at t=0
- \(u\) & \(d\) = Ratio of increase or decrease
- \(u = \frac{S_u}{S_0}\)
- \(d = \frac{S_d}{S_0}\)
For calls we’ll get a \(+\Delta\) & \(-B\). For puts we’ll get a \(-\Delta\) and \(+B\)
- Synthetic position
- Can use this formula to create an equivalent synthetic position
- Call payoff = purchasing \(+\Delta\) shares of stock, and borrowing \(-B\) to do so
- Put payoff = shorting \(-\Delta\) shares of stock, and lending \(+B\)
Arbitrage
- If the Option Prem \(\neq\) \(\Delta S + B\), then we buy what’s lower, and short what’s higher
- Profit = \(|C^* - C|e^{rt}\)
- Difference between what premium actually is and what it should be
- Profit is calculated at time t=t
Risk Neutral Pricing with Binomial Trees
Risk Neutral Pricing means to assume the stock’s expected return is equal to the risk free rate
This time we incorporate probabily that the stock rises/decreases between now and the next time interval
\(Prem = e^{-rt}[p(V_u)+(1-p)V_d]\)
- \(p = \frac{e^{(r-\delta)t}-d}{u-d}\)
- \(u=e^{(r-\delta)t+\sigma \sqrt{t}}\)
- \(d=e^{(r-\delta)t-\sigma \sqrt{t}}\)
- \(p = \frac{e^{(r-\delta)t}-d}{u-d}\)
p doens’t really equal the probability that the stock rises, but the risk-neutral probability, but you can just think of it this way in the formula
Price using multiple period tree
European Pricing
- Start with very right big side of the tree and find all payoffs there
- Then we sum up all of these payoff possibilities, times their probabilities (w combinatorics factor), and discount
- \(Prem = e^{-rt}[\sum \binom{m}{k} (prob_i)(payoff_i)]\)
- Combinatorics factor is the total number of periods (4 in this pic), choose either of the probability exponents
- \(p = \frac{e^{(r-\delta)h}-d}{u-d}\)
- \(u=e^{(r-\delta)h + \sigma \sqrt{h}}\)
- \(d=e^{(r-\delta)h - \sigma \sqrt{h}}\)
- We replace t w/ h which is our step size
- We can also workbackwards through each node like in American Options. B/c some problems might ask for prices at a particular time/node.
American Pricing
- American -> can exercise option at any time
- Key is to work backwards
- Find all payoffs at end period
- Work backwards step by step
- For each step back you take the max between:
- Immediate payoff if exercised immediately
- \(e^{-rh}[pC_u + (1-p)C_d]\)
- (^ = Pull back value = value that pulls you back from exercising early)
- For each step back you take the max between:
- Repeat until you work your way back to the first step
- The only reason to exercise an American Call early, is to start receiving dividends on the stock. So if there’s no dividends, then don’t exercise early.
- American Call price = European Call price when \(\delta = 0\)
- When it makes sense to exercise early:
- For American Calls:
- \(PV(Dividends) > PV(Interest~on~Strike) + PV(Premium)\)
- B/c you can start earning high dividend yields
- \(PV(Dividends) > PV(Interest~on~Strike) + PV(Premium)\)
- For American Puts:
- \(PV(Interest~on~Strike) > PV(Dividends) + PV(Premium)\)
- B/c you can start earning high interest on the strike
- \(PV(Interest~on~Strike) > PV(Dividends) + PV(Premium)\)
- For American Calls:
Options on Futures Contracts
Just use the Black Scholes framework, with the pre-paid forward equations. Instead of \(F^P(S)\) use \(F^P(F) = Fe^{-rt}\)
- Coaching actuaries teaches another very unnecessary method to calculate it:
Treat very similar to stocks, but you substitute F for S and r for \(\delta\) in the equations. And the formulas for \(\Delta\) and \(B\) for the replicating portfolio formulas are slightly different.
- \(V_0=\Delta F + B\)
- \(\Delta = \frac{V_u-V_d}{F (u_F-d_F)}\)
- (Change in Options Prices divided by Change in Futures Prices)
- \(u_F = e^{(r-r)h+\sigma \sqrt{h}}=e^{\sigma \sqrt{h}}\)
- \(d_F = e^{(r-r)h- \sigma \sqrt{h}}=e^{-\sigma \sqrt{h}}\)
- \(B=e^{-rh}[pV_u+(1-p)V_d]\)
- \(p=\frac{e^{(r-r)h}-d_F}{u_F-d_F}=\frac{1-d_F}{u_F-d_F}\)
- Just remember the regular formulas for u, d, and p, and substitute r for \(\delta\) and F for S.
- \(p=\frac{e^{(r-r)h}-d_F}{u_F-d_F}=\frac{1-d_F}{u_F-d_F}\)
- \(\Delta = \frac{V_u-V_d}{F (u_F-d_F)}\)
Then I’m pretty sure you have to use the American Option method, where you use this at every single node, and work your way backwards down the multiple period tree
Options on Currencies
- Just treat the foreign currency like a stock
- Exchange rate = \(x_0\) => \(S_0\)
- Domestic currency rfr = \(r_d\) => \(r\)
- Foreign currency rfr = \(r_f\) => \(\delta\)
Black Scholes
- This is how modern options are priced today.
- This framework assumes:
- Continuously compounded Stock Returns are Normally Distributed
- The ratios of the change of stock prices (\(\frac{S_t}{S_0}\)) are Lognormally distributed
- And thus, the log ratios of the change in stock prices (\(ln(\frac{S_t}{S_0})\)) are Normally distributed
- No Transaction costs
- Volatility and Risk free rate are constant
Risk Neutral Measure
- Assume \(\alpha = \gamma=r\)
- Expected return of stock and option are both equal to the rfr
- \(C = Se^{-\delta t}N(d_1) - Ke^{-rt}N(d_2)\)
- \(P = Ke^{-rt}N(-d_2) - Se^{-\delta t}N(-d_1)\)
- \(d_1 = \frac{log(\frac{S}{K}) + (r - \delta + .5 \sigma^2)t}{\sigma \sqrt{t}}\)
- \(d_2 = d_1 - \sigma \sqrt{t} = \frac{log(\frac{S}{K}) + (r - \delta - .5 \sigma^2)t}{\sigma \sqrt{t}}\)
- Just use 1st formula. Only difference in 2nd is the minus \(.5\sigma^2\)
- Derivation for Call Prem: (and put prem is similar)
- Call Prem = \(e^{-rt}E[Call ~ Payoff]\) = \(e^{-rt}[S_0e^{(r-\delta)t}N(d_1) - KN(d_2)]\) = \(S_0e^{-\delta t}N(d_1) - Ke^{-rt}N(d_2)\)
- When dealing w/ discrete dividends we can use prepaid fwd equations (these formulas also work w/ continuous divs)
- \(C = F^p(S)N(d_1) - F^p(K)N(d_2)\)
- \(P = F^p(K)N(-d_2) - F^p(S)N(-d_1)\)
- \(d_1 = \frac{log(\frac{F^p(S)}{F^p(K)}) + .5 \sigma^2 t}{\sigma \sqrt{t}}\)
- \(d_2 = d_1 - \sigma \sqrt{t}\)
- \(F^p(S) = S_0 - \sum div_i e^{-rt_i} = S_0 e^{-\delta t}\)
- Subtract the PV of each dividend from the stock price, or discount by \(\delta\)
- \(F^p(K) = Ke^{-rt}\)
- Just discount \(K\) by the rfr
True Pricing
- Start with all the equations above
- For the \(d_1\) & \(d_2\) equations, substitute \(\alpha\) for \(r\)
- For the \(C(K)\) & \(P(K)\) premium formulas, first compound the whole thing by \(r\), and then discount everything by \(\gamma\). The equation will turn into this:
- Call Prem = \(e^{-\gamma t}[S_0e^{(\gamma-\delta)t}N(d_1)-KN(d_2)]\)
- Use True Measure pricing by default
The GREEKS
- Formulas are in IFM packet so there’s no need to memorize. But, we do need to memorize/understand these last two columns
| GREEK | Partial Derivative | Explanation | Values/Magnitude |
|---|---|---|---|
| \(\Delta\) | \(\Delta = \frac{\partial V}{\partial S}\) | Change in Option price as stock increases by $1 | Higher Magnitude when the profit is higher. + for Calls, - for Puts |
| \(\Gamma\) | \(\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2}\) | Change in Delta as the stock increases by $1 | Always Positive for both calls and puts |
| Vega or \(\kappa\) | \(Vega = \frac{\partial V}{\partial \sigma}\) | Change in option price when volatility increases by 1% | High when stock and strike prices are close. Usually + |
| \(\theta\) | \(\theta = \frac{\partial V}{\partial t}\) | Change in option price when time moves forward one day closer to maturity | Usually - b/c options usually lose value w/ smaller time frame, but can be + if deep in the money |
| \(\rho\) | \(\rho = \frac{\partial V}{\partial r}\) | Change in option price when interest rate increases by 1% | Higher mag when in the money & T is longer. Calls = +, Puts - |
| \(\psi\) | \(\rho = \frac{\partial V}{\partial \delta}\) | Change in option price when dividend yield \(\delta\) increases by 1% | Higher mag when in the money & T is longer. Calls = -, Puts = + |
- Think of the Black Scholes main equation for most of these +/- changes (mostly for \(\theta\), \(\rho\), and \(\psi\))
Greeks of multiple positions
\(\Delta = \sum_{i=1}^{N} (n_i)(greek_i)\)
- For a portfolio of options, you just sum up all the \(\Delta\)’s or \(\Gamma\)’s or any Greek of each option, to get the net Greek value.
- When long, \(\Delta\) => 1. When Short => \(\Delta\) => -1.
Other Greek-ish formulas to memorize
Option Elasticity (\(\Omega\))
- \(\Omega = \frac{S\Delta}{V}\) = \(\frac{(V_t-V_0)/V_0}{(S_t-S_0)/S_0}\) = (% Change in Option price)/(% Change in Stock price)
- \(\Omega\) is the 3rd derivative of the options price, or derivative of \(\Gamma\)
- \(\Delta\) = Greek Delta of position (or net \(\Delta\) of portfolio)
- V = Option premium for Call or Put, (or net premium of portfolio)
- S = Current stock price
- Represents: If the stock changes by 1%, by what % does the option change?
- To remember: equation looks like SUD
Portfolio Elasticity
\(\Omega_{Portfolio} = \sum \omega_i \Omega_i\)
- \(\omega_i\) = % or weight of portfolio invested in option i
- \(\Omega_i\) = elasticity of option i
- So it’s just a weighted average of all the elasticities
- \(\Omega = \frac{S\Delta}{V}\) = \(\frac{(V_t-V_0)/V_0}{(S_t-S_0)/S_0}\) = (% Change in Option price)/(% Change in Stock price)
Option Volatility (\(\sigma_{option}\))
- \(\sigma_{option} = \sigma_{stock} |\Omega|\)
- Option volatility is higher than stock volatility
- \(\sigma_{option} = \sigma_{stock} |\Omega|\)
Sharpe Ratio (\(\phi\))
\(\phi = \frac{Risk~Premium}{Volatility}\)
- \(\phi_{Stock} = \frac{\alpha - r}{\sigma_{stock}}\)
- \(\phi_{Option} = \frac{\gamma - r}{\sigma_{option}} = \frac{\Omega}{|\Omega|} \phi_{stock}\)
- Last equation basically just means to take direction of \(\Omega\) (+/-) and multiply that by \(\phi_{stock}\)
We want the sharpe ratio to be high because we want a high return for low volatility
Risk Premium
- Risk Premium on a Stock = \(\alpha - r\)
- \(\alpha\) = Expected rate of return on a stock
- Risk Premium of an Option = \(\gamma - r = (\alpha - r)\Omega\)
- \(\gamma = \Omega\alpha + (1-\Omega)r\) = Expected return of an option
- Risk Premium on a Stock = \(\alpha - r\)
Delta Hedging
About
- Delta hedging might be the only thing actuaries actually use from IFM material
- Delta hedging is offsetting an options position by buying/selling \(\Delta\) shares of stock
- Market Maker is the person who sells the options contract
Delta Gamma Theta (\(\Delta\Gamma\theta\)) approximations
- \(C_{t+h}(S+\epsilon)\approx C_t(S) + \Delta\epsilon + .5\Gamma\epsilon^2 + \theta h\)
- A way to approximate the change in options price given the change in a stock price
- Only use part of the formula for however much they want you to approximate
Marking to Market
- Market maker adjusts his portfolio every day to stay hedged against market swings
- Best demonstrated with an example: Customer buys a t=91/365 day call option, so market maker sells the call option. Suppose S = $40, K = $40, \(\sigma\) = .3, \(r\) = .08, \(\delta\) = 0, and call contract is for 100 shares.
- Using Black Scholes framework and Greek Formulas we obtain these values in respect to the Call buyer:
- C(40) = 2.7804, \(\Delta\) = .5824, \(\Gamma\) = .0652, \(\theta\) = -.0173
- For the market maker, multiply each of these by -1 b/c we are selling the call
- C(40) = -2.7804, \(\Delta\) = -.5824, \(\Gamma\) = -.0652, \(\theta\) = .0173
- To hedge, the market maker buys .5824 shares of stock for every call
- Day 0:
- Market maker sells option contract so receives premium
- 2.7804*100 = 278.04
- Market maker buys \(\Delta\) shares for every option to hedge and spends:
- -.5824(100)(40) = -2329.6
- Market maker net position at t=0
- 278.04 - 2329.6 = -2051.56
- B/c it’s negative, we must borrow this money and will pay interest on it. If it were +, we would be lending this money and gaining interest
- Market maker sells option contract so receives premium
- Day 1: Stock rises to $40.50. Using Black Scholes w/ t=90/365, C(40) => $3.0621
- Market maker profit on shares he owns
- (40.50 - 40)(58.24) = 29.12
- Market maker profit on change in option value
- (2.7804 - 3.0621)(100) = -28.17
- Interest expense market maker must pay. We take previous period balance and compound by rfr
- \(-2051.56(e^{.08(\frac{1}{365})}-1) = -0.45\)
- Market Maker’s overnight profit:
- \(29.12 + -28.17 + -0.45 = 0.5\)
- Market maker must rebalance his portfolio
- Use Black Scholes to calculate new \(\Delta\) to be => .6142. So we must buy additional shares:
- (.6142 - .5824)(100) = 3.18
- So we must buy 3.18 more shares at new $40.50 price to rebalance portfolio: = 3.18(40.5) = 128.79
- This isn’t used in the profit equation for the day though
- Use Black Scholes to calculate new \(\Delta\) to be => .6142. So we must buy additional shares:
- Market maker’s new net position at Day 1:
- (3.0621)(100 options we sold) - (40.5)(61.42 shares we paid for) = -2181.3
- Market maker profit on shares he owns
- Day 2: Stock falls to $39.25, t=89/365, using Black Scholes C(40) => 2.3282
- Market maker profit on 61.42 shares he now owns
- (39.25 - 40.5)(61.42) = -76.775
- Market maker profit on option value change
- (3.0621 - 2.3282)(100) = 73.39
- Market maker pays interest on previous loan balance
- \(-2181.3(e^{.08/365}-1) = -0.48\)
- Market maker’s overnight profit:
- \(-76.775 + 73.39 + -0.48 = -3.865\)
- Market maker profit on 61.42 shares he now owns
- Here’s a summary table of what happens each day:
- Using Black Scholes framework and Greek Formulas we obtain these values in respect to the Call buyer:
- Steps in Marking to Market
- Find market maker’s net Delta position, and offset by buying/selling that many shares
- Find market maker’s net position from shares and option premiums
- For the next day
- Use new Stock price and t in the Black Scholes framework to find new Delta and Call premium
- Find the profits for your stock and option premium positions
- Find profit for how much you pay/receive in interest from previous position
- Add/subtract shares from portfolio to mark to market (rebalance portfolio) w/ new Delta
- Find new Net position
- Remember, interest profit never gets added to the net portfolio value
Gamma Hedging
- To stay even more hedged, not only do we make our net \(\Delta\) of our position = 0, but we also make our net \(\Gamma\) of our position = 0.
- We do this by buying/selling shares of another call option to get a net \(\Gamma\) of 0, but then have to buy/sell stock shares to get a net \(\Delta\) of 0
- We have 2 options w/ strikes: \(K_1\) and \(K_2\), each w/ their own \(\Delta\)’s and \(\Gamma\)’s
- If we sell the \(C(K_1)\) option, we \(\Gamma\) hedge by buying \(\frac{\Gamma_1}{\Gamma_2}\) \(K_2\) Call options to make our net \(\Gamma\) 0
- Our combined \(\Delta\) becomes:
- \(\Delta_{combined} = -\Delta_1+\frac{\Gamma_1}{\Gamma_2}\Delta_2\)
- You can logic your way to this equation if you just remember to buy \(\frac{\Gamma_1}{\Gamma_2}\) contracts of option 2
- So if this is negative we purchase that many shares to offset our position
Exotic Options
Asian Options
- Deals with the average price of the stock
- 2 different kinds of averages
- Arithmetic Average A(T) = just regular mean
- Geometric Mean G(T) = \((S_1*S_2*S_3*S_4*...)^{1/N}\)
- Do not use the initial stock price in calculating the averages, only future ones
- Some notes about the averages:
- Price of Asian Option \(\leq\) price of equivalent European option
- To remember this, assume that if you buy an option you know which direction the stock is moving, and the end price will be better than the average.
- G(T) \(\leq\) A(T)
- More sampling -> Lower volatility -> Decreases the average
- So daily stock prices will have a lower average than weekly stock prices
- Price of Asian Option \(\leq\) price of equivalent European option
- Recall that payoffs are calculated:
- Call payoff = Max{0, S-K}
- Put payoff = Max{0, K-S}
- For Asian options, we replace either S, or K using either A(T) or G(T)
- Between Call/Put, S/K, A(T)/G(T) there are 8 diff combinations
- Probs will say “avg price” for replacing S, or “avg strike” for replacing K
- Probs often have you construct a binomial tree, calculate payoffs at the end of each node, and multiply by the probability of each end node to price the option
Barrier Options
Types:
- Knock-In/Out (Up and In / Down and Out) = Option goes in/out of existence if at any point between time 0 and T the stock crosses the barrier once
- Rebate option: Pays a fixed amount if the barrier is reached
- Down Rebate = If barrier is below the current price
- Up Rebate = If barrier is above the current price
- Pricing
- On IFM, may have to build a binomial tree to solve problems
- In real life, these are priced using computer simulations
- Barrier Price \(\leq\) Ordinary Option
- b/c more restrictive
- Parity relationship for option premiums
- Knock-in + Knock-out = Ordinary option
Compound Options
- Option to buy an Option
- Have 2 strikes and 2 expirations
- \(t_0\) is current time. At \(t_1\), you have the option to buy (for a price of \(\$x\)) a European option with a strike price of \(\$K\). At time \(T\), the underlying European option expires.
- \(S^*\) is the critical value that if the stock rises above this value, then it makes sense to exercise the compound option.
Compound Option Parity
- CallonCall - PutonCall = Call - \(xe^{-rt_1}\)
- CallonPut - PutonPut = Put - \(xe^{-rt_1}\)
- Problems may have you use Black Scholes to value the underlying option at time t
- For profits, make sure you discount all the cash flows to t=0
- How much you paid for the compound option at t=0
- How much you’ll earn/spend at t=t when you exercise the compound option to buy/sell the underlying option
- The payoff of the underlying option at t=T
Gap Options
- Gap options have a regular strike price \(K_1\) that you buy/sell the asset for, but you can only do so if the stock is more/less than the “trigger” price \(K_2\) at expiration
- Use the same Black Scholes formulas except we change the \(d_1\) formula to have \(K_2\) in place of \(K_1\)
- \(C = Se^{-\delta t}N(d_1)-K_1e^{-rt}N(d_2)\)
- \(P = K_1e^{-rt}N(-d_2) - Se^{-\delta t}N(-d_1)\)
- \(d_1 = \frac{log(\frac{Se^{-\delta t}}{K_2e^{-rt}}) + .5\sigma^2t}{\sigma\sqrt{t}} = \frac{log(\frac{S_0}{K_2})+(r-\delta+.5\sigma^2)T}{\sigma \sqrt{T}}\)
- \(d_2 = d_1 - \sigma\sqrt{t}\)
- Put-call parity for gap options:
- GapCall - GapPut = \(S_0 e^{-\delta T} - K_1 e^{-rT}\)
- Basically everything is the same for Gap options except that we use the barrier/gap \(K_2\) in the \(d\) equations
- Gap options are unique in that they are actually not optional. They must be exercised if the stock price moves and stays past the trigger price. Thus a negative payoff and negative premiums are possible.
Exchange Options
- AKA Outperformance Options
- Basically an option to exchange an asset with another
- S is the price of risky asset 1, K is the price of risky asset 2
- We are given dividend yields (\(\delta_S\) & \(\delta_K\)), volatilites (\(\sigma_S\) & \(\sigma_K\)), and correlation between the 2 continuously compounded dividend yields (\(\rho\))
- Call option is giving asset K (2), and receiving asset S (1). (Paying strike K, receiving stock S) (Same as normal). Put is giving asset S and receiving K.
- Call Prem = \(Se^{-\delta t}N(d_1) - Ke^{-rt}N(d_2)\)
- \(d_1 = \frac{log(\frac{Se^{-\delta_S t}}{Ke^{-\delta_K t}}) + .5\sigma^2t}{\sigma\sqrt{t}}\)
- \(d_2 = d_1 - \sigma\sqrt{t}\)
- \(\sigma = \sqrt{\sigma^2_S + \sigma^2_K -2\rho\sigma_S \sigma_K}\)
Manipulating payoffs to look like exchange/regular options
- Exam problems may give you a lot of info, and you narrow it down to the Max(X(t), Y(t))
- Rearrange to look like: Y(t) + Max(X(t) - Y(t), 0)
- This is like a call option, where you pay Y and receive X, + a prepaid fwd of stock Y (so discount it by the rfr).
3 manipulations
- Max(A, B) = A + Max(B - A, 0)
- Call option, pay A and receive B
- Max(cA, cB) =
- c*Max(A, B) if c > 0
- c*Min(A, B) if c < 0
- Max(A, B) + Min(A, B) = A + B
Even more exotic options
Forward Start Option
- Prepaid forward on an option. You pay now, and receive an option in the future, and strike price is unknown until you receive the option, but it will be a %age of the stock
- Pricing
- Set the strike price: \(K = xS_t\) (x% of the stock price). Set the Time: \(t = t - T\). Finally, discount the option price back to t=0 by the rfr.
- Call = \(S_t e^{-\delta (T-t)}N(d_1) - xS_t e^{-r(T-t)}N(d_2)\)
- \(d_1\) = same as usual, substitute \(xS_t\) for K, and T-t for t
- Times
- t=0: when prepaid fwd contract begins
- t=t = when underlying option is delivered
- t=T = when underlying option expires
Chooser Option
- Owner gets to choose whether the option becomes a call or a put at t=t, strike (K) and expiration date (T) stay the same
- \(t_0\) = today, \(t_t\) = time when option owner gets to choose call/put, \(t_T\) = time when option expires
- \(V_t = Max(C_{T-t}, P_{T-t})\)
- \(V_t\) = value of option at time t
- Rearranging using put-call parity and discounting by dividend yield:
- \(V_0 = e^{-\delta (T-t)}C(S_0 ,Ke^{-(r-\delta)(T-t)}, t) + P(S_0 , K, T)\)
- Hopefully this is so rare to see on the exam
Lookback Option
Depends on the Min or Max price of the stock over the time period
4 diff types:
| Type | Explanation | Payoff |
|---|---|---|
| Standard Lookback Call | You get to buy the stock for the lowest possible price over the period | \(S_T-Min(S)\) |
| Standard Lookback Put | You get to sell the stock for the higest possible price over the period | \(Max(S)-S_T\) |
| Extrema Lookback Call | You get to buy the stock for K, and sell it for the best possible price over the period | \(Max(Max(S)-K,0)\) |
| Extrema Lookback Put | You get to sell the stock for K and buy it for the best possible price over the period | \(Max(K-Min(S),0)\) |
- Way to remember:
- Standard: The STRIKE price is the moving piece that gets locked in at the best price.
- Extrema: The STOCK price is the moving piece that gets locked in at the best price.
Properties of Lognormal
- \(\frac{S_t}{S_0} \tilde{} LogN(m,v^2)\)
- A stocks return follows a normal distribution
\(ln(S_t) \tilde{} N(m = ln(S_0)+(\alpha-\delta-.5\sigma^2)t, v^2 = \sigma^2 t)\)
- \(ln(\frac{S_t}{S_0}) \tilde{} N(m = (\alpha -\delta - .5\sigma^2)t, v^2 = \sigma^2 t)\)
- A stock’s continuously compounded return is normally distributed
- \(S_t = S_0 e^{(\alpha-\delta-.5\sigma^2)t + \sigma \sqrt{t} *Z}\)
- Where \(Z \tilde{} N(0,1)\)
- Coaching Actuaries says we should memorize this
Following Lognormality, we get these equations:
\(E[S_t]=S_0e^{(\alpha-\delta)t}\)
\(Var(S_t) = E(S_t)^2(e^{\sigma^2 t}-1)\)
\(Med[S_t]=S_0e^{(\alpha-\delta-.5\sigma^2)t}=E[S_t]e^{-.5\sigma^2 t}\)
- \(E[S_t|S_t>K]=S_0e^{(\alpha-\delta)t}\frac{N(d_1)}{N(d_2)}\)
- \(E[S_t|S_t<K]=S_0e^{(\alpha-\delta)t}\frac{N(-d_1)}{N(-d_2)}\)
- \(P(S>K)=N(d_2)\)
- \(P(S<K)=N(-d_2)\)
- \(d_1 = \frac{log(\frac{S_0}{K})+(\alpha-\delta+.5\sigma^2)t}{\sigma\sqrt{t}}\)
- Assuming no arbitrage \(\alpha = r\)
- \(d_2 = d_1 - \sigma\sqrt{t} = \frac{log(\frac{S_0}{K})+(r-\delta-.5\sigma^2)t}{\sigma\sqrt{t}}\)
- Only diff is we subtract \(.5\sigma^2\) in the numerator
- \(d_1 = \frac{log(\frac{S_0}{K})+(\alpha-\delta+.5\sigma^2)t}{\sigma\sqrt{t}}\)
Fitting Stockprices to a Lognormal Distribution
Exam might give you a table of stock prices over time, and you’ll need to compute parameters
Historical Volatility (\(\sigma\)) is calculated in 3 steps
- Find log ratios
- \(log(\frac{S_t}{S_{t-h}})\)
- Find unbiased estimator of the variance of these ratios
- \(S^2 = \frac{1}{n-1}\sum(x_i-\bar{x})^2\)
- Or just plug values into multiview
- Multiply by 1/h and square root to get SD
- \(\sigma = \sqrt{S^2/h}\)
- (For weekly values, h = 1/52, so we’d multiply it by 52)
- \(\sigma = \sqrt{S^2/h}\)
- Find log ratios
Volatility on a stock (another way to get it if they give you a formula)
\(\sigma = \sqrt{\frac{Var(Ln(S_t))}{t}}\)
- Ex: They give that: \(Var(Ln(F^P_{t,0.5}(S)))=.09t\)
- \(\sigma = \sqrt{\frac{.09t}{t}}=\sqrt{.09}=.3\)
- Computing \(\hat{\alpha}\), (Continuously compounded expected return)
- Recall that \(ln(\frac{S_{t+h}}{S_t}) \tilde{} N(m = (\alpha - \delta - .5\sigma^2)h, v^2 = \sigma^2 h)\)
- Once you find all the log ratios, compute the average log ratio (\(\bar{x}\) from the multiview), and this will give you \(\bar{r}\) which equals \(m\) in the above.
- Use the steps above to compute the sample volatility \(\hat{\sigma}\) using those log ratios
- For \(h\), use the step size, so if you have monthly stock prices: multiply by \(h=\frac{1}{12}\)
- (Note, this is different from computing \(\hat{\sigma}\), where we divide by \(\frac{1}{h}\), so we end up multiplying by 12)
- Once you have those pieces, you can solve for \(\alpha\), but this will be \(\hat\alpha\) b/c we are using historical prices and thus is an estimate.
- Recall that \(ln(\frac{S_{t+h}}{S_t}) \tilde{} N(m = (\alpha - \delta - .5\sigma^2)h, v^2 = \sigma^2 h)\)
Variance/Covariance/Correlation Review
- \(Var(X) = E[(X-E(X))^2] = E(X^2)-E(X)^2\)
- \(S^2 = \frac{\sum (x_i -\bar{x})^2}{N-1}\)
- Unbiased estimator of the variance. Use when dealing w/ a sample
- \(\sigma^2 = \frac{\sum (x_i -\bar{x})^2}{N}\)
- Actual variance. Use when the population mean is known
- \(Var(aX+bY) = a^2Var(X)+b^2Var(Y)+2abCov(X,Y)\)
- Cov = 0, when X and Y are independent
- Cov is usually given on the exam
- \(Var(X+Y+Z) = Var(X)+Var(Y)+Var(Z)+2Cov(X,Y)+2Cov(X,Z)+2Cov(Y,Z)\)
- Variance of each, plus covariance of every combination with combinatorics factor (2)
- \(Cov(X,Y)=\frac{\sum (x_i -\bar{x})(y_i -\bar{y})}{N-1}\)
- This is the unbiased estimator, use only N in the denominator for actual covariance
- \(Cov(X,Y) = E(XY)-E(X)E(Y)\)
- \(Cov[aX+bY,cP+dQ]=Cov[aX,cP]+Cov[aX,dQ]+Cov[bY,cP]+Cov[bY,dQ]=acCov[X,P]+adCov[X,Q]+bcCov[Y,P]+bdCov[Y,Q]\)
- \(\rho=Corr(X,Y)=\frac{Cov(X,Y)}{SD(X)SD(Y)}\)
Actuarial Specific Risk Management
- For the exam: You need to understand these insurance products, how they resemble traditional options, and how to hedge the risk.
| Insurance Product | Explanation | How it’s like an option (embedded option) |
|---|---|---|
| GMDB: Guaranteed Minimum Death Benefit | Life Insurance. Get a min amt if insured dies | Max(Account Value, Original Amt Invested) = Max(\(S_T, K\)) = \(S_T + Max(K - S_T, 0)\) So this is like a Euro Put option |
| GMAD: Guaranteed Min Accumulation Benefit (AKA guar min Maturity benefit) | Guarantees a min value for the account, if you live that long | Max(Account value, Guar Min Benefit) = Max(\(S_T, K\)) = \(S_T\) + Max(\(K-S_T\), 0) = Euro Put |
| GMWB: Guaranteed Min Withdrawal Benefit | Get a guaranteed min withdrawal amt over a specified period of time if you live long enough | |
| GMIB: Guaranteed Min Income Benefit | Guarantees a fixed purchase price of an annuity at some future time. | |
| Earnings Enhanced Death Benefit | Pays at death, only if the acct. value is > Original amt. Pays based based on amt of increase | Receive (40%)*Max(\(S_T - K, 0\)). Like a %age of a Euro Put |
| Mortgage Guarantee Insurance | Lender (bank) buys this. Pays off mortgage if borrower defaults. After default, lender sells property at an auction, and insurance pays the leftover amt and settlement costs (taxes, maintenance, missed pmts, etc.) These are usually packaged up and sold as MBS’s. | Payoff = Max(B + C - R, 0) B = Outstanding loan balance C = Settlement Costs R = Net amt received at foreclosure Like a Put option (B+C is the static part) |
| Guaranteed Replacement Cost Coverage | Purchased by consumer, added to regular mortgage insurance. Say your $300k house burns down, but costs $400 to repair, regular mortgage ins pays $300k, thte GRCC pays $100k | Payoff of regular property ins = Min(C, I) Payoff of GRCC = Max(0, C-I) C = Cost of replacement I = Amt of base policy Like a call option |
| Inflation Indexing | Pension pmts are adjusted to keep up w/ inflation. Full Indexing: Purchasing power of each pmt \(\geq\) purchasing power of original pmt Partial Indexing: Purchasing power of each pmt \(\geq\) Some defined amt |
See below |
Inflation Indexing
- Full Indexing:
- \(P_t = Max(P_0 \frac{I_t}{I_0}, P_{t-1})\)
- \(I_t\) = CPI at t=t
- \(P_t\) = Pension pmt at t=t
- First part ensures that the pension pmts will rise when inflation rises
- 2nd part ensures that the pension pmts won’t decrease even in deflation
- Making this look like options:
- \(P_1 = P_0 + P_0 Max(\frac{I_1}{I_0}-1, 0)\)
- Initial Pmt + Call Option
- \(P_1 = P_0 (\frac{I_1}{I_0}) + P_0 Max(0, 1 - \frac{I_1}{I_0})\)
- Indexed Initial Pmt + Put Option
- \(P_1 = P_0 + P_0 Max(\frac{I_1}{I_0}-1, 0)\)
Using derivatives to manage risk in insurance and annuity products
Say you write a GMAB variable annuity that’s closely linked to the S&P 500. So you are basically writing puts, you may want to hedge by purchasing puts w/ similar strike prices and expiration dates to offset this risk.
Dynamic Hedging = Frequent buying/selling derivatives to match the value of the guarantees you sold. You’ll need to know/estimate the greeks to stay hedged.
Static Hedge (AKA Hedge and Forget) = No need to keep buying/selling derivatives, you can just leave it as is and stay hedged.
Hedging of Catastrophy Risk
- Usually insureds suffer independent catastrophes. But some catastrophes aren’t independent (floods, giant forest fires)
- You can hedge by buying reinsurance, or buying weather derivatives and catastrophe bonds.
Corporate Finance
Risk and Return
Return
- Actual return of a portfolio
- \(R_p=\sum x_i R_i\)
- \(x_i\) = portfolio weight of asset i
- \(R_i\) = return from asset i
- \(R_p=\sum x_i R_i\)
- Expected return of a portfolio
- \(E[R_p]=\sum x_i E[R_i]\)
- Average Annual Return (default average)
- \(\bar{R} = \frac{\sum R_t}{T}\)
- Expected return based on past performance
- Compund annual return = geometric avg
- \(\bar{R} = [(1+R_1)(1+R_2)...(1+R_T)]^{(1/T)}-1\)
- IRR = Internal Rate of Return
- \(\bar{R} = \frac{Final ~ Value}{Initial ~ Value}^{(1/T)}-1\)
- \(S_t=S_0(1+\bar{R})^T\)
- Just use this second more intuitive equation and solve for \(\bar{R}\)
- Historical Return
- \(R_{t+1}=\frac{Div_{t+1}+P_{t+1}}{P_t}-1\)
- \(R\): Return
- \(Div_t\): dividend amount received at t=t
- \(P_{t}\) Price at t=t
- Basically this is just how much you have at the end, divided by how much you started w/, -1 to get the return.
- \(R_{t+1}=\frac{Div_{t+1}+P_{t+1}}{P_t}-1\)
- Quarterly -> Annual returns
- \(1+R_{annual}=(1+R_{Q1})(1+R_{Q2})(1+R_{Q3})(1+R_{Q4})\)
- These can be any time interval, as long as they all add up to 1 year. Can even combine weeks and months
- Standard Error of the return = \(\frac{SD(R_i)}{\sqrt{N}}\)
- Usually, when we calculate volatility, we use the log ratios of the returns. This time, just use the regular returns, calculate teh SD of those returns (Using the sample SD formula (/N-1)), then finally divide by the sqrt of the number of obs (1/N) at the end.
- XX% Confidence Interval for the expected return:
- = \(\bar{R} \pm (z^*) \frac{SD(R_i)}{\sqrt{N}}\) = Historical Return \(\pm (z~score*Standard~Error)\)
Risk
- Systematic Risk: (Common Risk), (Undiversifiable Risk), (Market Risk)
- This risk hits everyone, can’t be diversified
- Ex: Terrorist attacks, economic cycles, interest rates, widespread natural disasters
- Non Systematic Risk: (Independent Risk), (Firm specific risk), (idiosyncratic risk), (unique risk), (unsystematic risk), (diversifiable risk)
- This risk can be avoided by diversifying
- Ex: scandal at 1 firm, airline crash, drug failing trial
Variance and Volatility formulas
- Main Equation for the variance of a portfolio of returns
- \(Var(R_p)=\sum \sum x_i x_j Cov(R_i, R_j)\)
- You do this for every combination of returns, w/ combinatorics factors
- For ex, this is for a portfolio of 3 stocks
- \(Var = 2x_1x_2Cov(R_1, R_2) + 2x_1x_3Cov(R_1, R_3) + 2x_2x_3Cov(R_2, R_3) +\) \(x_1^2Cov(R_1, R_1) + x_2^2Cov(R_2, R_2) + x_3^2Cov(R_3, R_3)\)
- And recall that the Cov() of a return w/ itself is just the variance
- \(Var = 2x_1x_2Cov(R_1, R_2) + 2x_1x_3Cov(R_1, R_3) + 2x_2x_3Cov(R_2, R_3) +\) \(x_1^2Cov(R_1, R_1) + x_2^2Cov(R_2, R_2) + x_3^2Cov(R_3, R_3)\)
- \(Var(R_p)=\sum \sum x_i x_j Cov(R_i, R_j)\)
- Variance of an equally weighted n-stock portfolio (shortcut)
- \(Var(R_p) = \frac{1}{n}(Avg~Var) + (1-\frac{1}{n})(Avg~Cov)\)
- If the stocks are “independent with identical risks”, then the avg covariance will be 0
- \(Var(R_p) = \frac{1}{n}(Avg~Var) + (1-\frac{1}{n})(Avg~Cov)\)
- Volatility of a portfolio of n-stocks with covariance and correlation of 0 with each other, and have identical risks
- Volatility = \(\frac{SD(Individual~Risk)}{\sqrt{n}}\)
- Can be derived from the above equation
- Volatility = \(\frac{SD(Individual~Risk)}{\sqrt{n}}\)
- Covariance and Correlation formulas
- \(Cov(R_i, R_j)=E[(R_i-E[R_i])(R_j-E[R_j])] = \frac{1}{T-1} \sum (R_{i,t} - \bar{R_i})(R_{j,t} - \bar{R_j})\)
- \(Corr(R_i, R_j)=\frac{Cov(R_i, R_j)}{SD(R_i)SD(R_j)}\)
- General Portfolio Volatility:
- \(\sigma_P = \sum \rho_{i,P} \sigma_i x_i\)
- \(\sigma_P\) = volatility of portfolio
- \(x_i\) = weight of stock i in the portfolio
- \(\sigma_i\) volatility of stock i
- \(\rho_{i,P}\) = correlation of stock i w/ portfolio
- Remember “Op Rox”
- \(\sigma_P = \sum \rho_{i,P} \sigma_i x_i\)
Mean Variance Portfolio Theory
M-V Portfolio Theory = Says that investors can evaluate the risk/return of an investment based on expected returns. We use this to find the optimal allocation of assets.
- Assumptions:
- All investors are risk averse (prefer less risk for return)
- Expected return, variances, and covariances of all assets are known
- To determine the optimal portfolio, we only need to know the above 3 measures
- There are no transaction costs or taxes
- Assumptions:
Portfolio of 2 Risky Assets (No risk free assets)
- Anywhere along the blue line is the “efficient frontier” meaning we get enough return for the given level of risk
- MVPT says that all investors will invest along the efficient frontier somewhere depending on their risk preference
- The 2nd plot shows what happens if we include short sales (shorting 1 and investing in the other)
Correlation effect
- Correlation will range between -1 and 1, so our efficient frontier curve will look something like one of these depending on the correlation.
- Adding even more stocks that aren’t perfectly correlated yields a superior efficient frontier with this added diversification
Effect of adding t-bills to the portfolio
- Do this to reduce some risk
- Expected return with adding T bills to portfolio
- \(E[R_{xp}] = xE[R_p] + (1-x)r_f\)
- x = %age of portfolio that consissts of stocks
- \(E[R_{xp}] = xE[R_p] + (1-x)r_f\)
- Standard Deviation
- \(SD(R_{xp}) = xSD(R_p)\)
- Standard deviation of a portfolio with t bills, is just the %age of stocks in portfolio, times the SD of the stock portfolio. B/c t-bills have no volatility
- \(SD(R_{xp}) = xSD(R_p)\)
Capital Allocation Line (CAL)
- Rearrange the expected return of a portfolio:
- \(E[R_{xp}]=xE[R_p]+(1-x)r_f\)
- Start w/ the above, and rearrange to:
- \(E[R_{xp}]=r_f + \sigma_{xp} \frac{E[R_p]-r_f}{\sigma_p}\)
- \(p\) = 100% weight invested in mkt portfolio
- \(xp\) = x% is invested in mkt portfolio
- \(E[R_{xp}]=xE[R_p]+(1-x)r_f\)
- Graph of the CAL:
- Notes on the CAL:
- Intercept: \(r_f\). Slope: Sharpe ratio of the portfolio
- The Y-int represents a portfolio of 100% rf assets, at the far right this represents a portfolio of 100% risky assets
- We can short risk free assets and invest more in risky assets thus extending the line (buying stocks on margin, AKA leveraged portfolio)
Tangent Portfolio
- Combine the CAL and the efficent frontier to find the optimal portfolio
- Where the CAL is tangent to the efficent frontier, is the optimal efficient portfolio
- So the most efficient portfolio on the efficient frontier, is where the slope equals the sharpe ratio
- So you basically start w/ a portfolio of stocks P and build an efficient frontier, the most efficient combination is where the slope equals the sharpe ratio. THEN, we can move along the CAL depending on how much we want to invest in risk free assets.
MV portfolio theory says that all investors have the exact same efficient frontier, and exact same rfr. So thus they all have the same optimal portfolio. And that they should just move along the CAL based on how much risk they really want by adjusting the amount invested in risk free assets.
- Capital Market Line (CML)
- When we use the entire market portfolio as the “Optimal Risky Portfolio”, then we refer to the CAL as the Capital Market Line (CML)
Should a new investment be added to the portfolio?
If the following is true, then we should add the asset:
\(\frac{E[R_{New}]-r_f}{\sigma_{New}} > \rho_{New, P} \frac{E[R_P]-r_f}{\sigma_P}\)
- (Sharpe Ratio new asset) > (Correlation) (Sharpe Ratio existing portfolio)
Capital Asset Pricing Model (CAPM)
- \(r_i = E[R_i] = r_f + \beta_i(E[R_{mkt}-r_f)\)
- \(r_i\) = Required return
- \(E[R_i]\) = Expected return
- \(E[R_{mkt}]-r_f\) = Mkt Risk premium, AKA Expected excess return of the mkt
- \(\beta_i(E[R_{mkt}-r_f)\) = Risk premium for security i, AKA expected excess return of security i
- Notice how we set the required return equal to the expected return. The CAPM is really used to find the required return of asset i, but if the market portfolio is efficient, then this will also be the expected return of asset i.
Beta
- Beta = Asset’s sensitivity to Market (systematic risk) = The % change when the mkt changes by +1%
- Several ways to calculate:
- \(\beta= \frac{\Delta Asset}{\Delta Mkt} =\frac{Change ~ in ~ Asset's ~ Return}{Change ~ in ~ Market's ~ Return}\)
- \(\beta = \frac{Cov(R_i, R_{mkt})}{\sigma^2_{mkt}}\) = (Covariance of Asset w/ Mkt) / (Variance of the mkt return)
- (\(\beta=\frac{cov}{\sigma^2_M}\)) = “Bomb2 Cov”
- Which can be rearranged to:
- \(\beta = \rho_{i, mkt} \frac{\sigma_i}{\sigma_{mkt}}\) = corr * (SD asset / SD mkt)
- Beta can also be calculated with regression:
- x-axis: Mkt excess return
- y-axis: Asset excess return
- Beta = regression line slope
- Ex: The mkt swings between -30% to 50%, and 1 asset swings between -25% and 30% when the mkt is weak/strong.
- Then Beta = \(\frac{30 - (-25)}{50-(-30)}\) = .6875
- Beta of a Portfolio
- \(\beta_P = \sum x_i \beta_i\)
- Several ways to calculate:
CAPM Assumptions:
- No Friction (no taxes, no transaction costs, investors can buy/lend at the rfr, investors can buy/sell securities at competitive prices)
- Investors hold only efficient portfolios (greatest return for a given volatility). Meaning investors are risk averse, rational, and utility maximizing
- Investors have Homogeneous expectations regarding volatilities, expected returns, correlations of securities. So the optimal risky portfolio is the same for everyone
Security Market Line (SML) = Graphical representation of the CAPM
- Slope = Mkt risk premium
SML and Alpha
- The required return might not be equal to the expected return
- \(\alpha_i = E[R_i] - r_i\)
- Difference between the expected return and the required return
- Based on the CAPM, alpha for all stocks is 0, all stocks lie on the SML and there is no difference between the expected and required return
- \(\alpha_i = E[R_i] - r_i\)
- Portfolio improvement
- If the stock has a positive alpha (if the expected return is higher than the required return calculated in the CAPM), then we should add it to our portfolio.
Cost of Capital
Definitions
- Cost of Capital = Rate of return that providers of capital require in order for them to contribute money to the firm
- Cost of Equity = Rate of return that equity holders require in order for them to contribute money to the firm
- AKA: Equity cost of capital, required return on equity
- Cost of Debt = RAte of return that debt holders require in order for them to contribute money to the firm
- AKA: Debt cost of capital, required return on debt
Quick Summary
- Unlevered Cost of Capital = Asset cost of Capital
- Use the WACC w/o the corp tax rate discount
- Levered Cost of Capital
- Use the WACC with the corp tax rate discount
- Equity Cost of Capital
- Use CAPM
- Or use MM prop 2 formula
- Debt cost of Capital
- \(r_d=y-pL\)
- Or us CAPM with Debt beta
Unlevered Cost of Capital, AKA Asset Cost of Capital
- For valuing a project that is financed by equity only, without any debt.
Usually have to use a firm w/ some debt as a comparable, so we use the WACC equation w/o the corp tax rate part
- \(r_U = w_E r_E + w_D r_D\)
- \(r_U\) = Asset or Unlevered Cost of Capital
- \(r_E\) = Equity cost of capital
- \(r_D\) = Return on debt
- \(w_E = \frac{E}{E+D}\) = Weight of Equity
- \(w_D = \frac{D}{E+D}\) = Weight of Debt
- \(E\) = Amt of equity
- \(D\) = Net Debt = Debt - Cash
- Unlevered (Asset) Beta
- \(\beta_U = w_E \beta_E + w_D \beta_D\)
Levered Cost of Capital
For valuing a project that is financed by a combination of equity and debt.
- Weighted Average Cost of Capital (WACC)
- \(r_{WACC} = w_E r_E + w_D r_D (1-\tau_C)\)
- \(r_{WACC}\) = WACC required return
- \(r_E\) = Equity cost of capital
- \(r_D\) = Return on debt
- \(w_E=\frac{E}{E+D}\) = Weight of Equity
- \(w_D=\frac{D}{E+D}\) = Weight of Debt
- WACC relation to the unlevered return:
- \(r_{WACC}=r_U - w_D r_D \tau_C\)
- (WACC required return) = (Unlevered required return) - (Weight of debt)(Debt required return)(Corp tax rate)
- \(r_{WACC}=r_U - w_D r_D \tau_C\)
- \(r_{WACC} = w_E r_E + w_D r_D (1-\tau_C)\)
Return on Equity (Equity cost of capital)
- Calc using CAPM:
- \(r_E = r_f + \beta(E[R_{mkt}] - r_f)\)
- MM Proposition 2 also gives us this equation:
- \(r_E=r_U+\frac{D}{E}(r_U-r_D)\)
- \(r_E\) = Cost of capital of levered equity
- \(r_U\) = Unlevered Equity Cost of Capital when the firm has no debt
- \(\frac{D}{E}\) = Firm’s Debt to Equity ratio
- \(r_D\) = Required return on debt
- Remember: “Re Rude Rurd” … seems weird but helps me
- \(r_E=r_U+\frac{D}{E}(r_U-r_D)\)
Debt Cost of Capital
- This is the required return investors expect when lending
- 2 Methods to calculate:
- Expected return of a bond:
- \(r_d = y - pL\)
- Return on Debt = (Yield to Maturity of Bond) - (Prob of default)*(Expected Loss Rate (expected loss per $1 of debt))
- \(r_d = y - pL\)
- CAPM using debt betas:
- \(r_d = r_f + \beta_d (Mkt~Risk~Premium)\)
- Return on Debt = (rfr) + (Debt Beta)*(Mkt risk prem)
- \(\beta_d\) = The \(\beta\) used here is diff than our equity \(\beta\). Risker bonds have a higher \(\beta\), so we look it up on a table based on it’s rating (AAA, A, BBB, BB, …)
- \(r_d = r_f + \beta_d (Mkt~Risk~Premium)\)
- Expected return of a bond:
- Effective After-tax Cost of Debt = \(r_D(1-\tau_C)\)
- \(r_D\) = Return on debt
- \(\tau_C\) = Corporate tax rate
- When a company pays interest on debt, this amt is tax deductible for the firm. This reduces the firm’s debt cost of capital b/c it’s less costly.
Value of a Firm
- \(V = D + E\)
- Value of a firm = Debt + Equity
Value of a Levered vs Unlevered firm
- \(V_L = \sum \frac{Cash~Flow_i}{(1+r_{wacc})^i}\)
\(V_U=\sum \frac{Cash~Flow_i}{(1+r_U)^i}\)
- \(V_L = V_U + D \tau_C\)
- Value of a levered firm = value of an unlevered firm + Debt * (Corp Tax Rate)
- This Debt * \(\tau\) is also known as the PV of the interest tax shield
- \(V_U = \frac{FCF}{r_U}\)
- Value of an unlevered firm = (Free Cash Flow) / (Unlevered return)
- \(D \tau_C\) = PV of the interest rax shield
- = (Amt of Debt)(Corp tax rate)
- Value of a levered firm = value of an unlevered firm + Debt * (Corp Tax Rate)
Enterprise Value
- Enterprise Value = Firm’s underlying business value separate from cash holdings
- Enterprise Value = Mkt Cap + Net Debt
- Net Debt = Debt - Cash (excess cash and short term investments)
- Beta of firm’s underlying enterprise
- \(\frac{E}{E+D-C} \beta_E + \frac{D}{E+D-C} \beta_D + \frac{C}{E+D-C} \beta_C\)
- E = Amount of equity orr “Market Cap”
- D = Net Debt
- C = Cash and short term investments
- \(\frac{E}{E+D-C} \beta_E + \frac{D}{E+D-C} \beta_D + \frac{C}{E+D-C} \beta_C\)
- Enterprise Value = Mkt Cap + Net Debt
Arbitrage Pricing Theory (APT)
- A multi-factor model. Considers a collection of well diversified portfolios to find the expected return of an asset.
- Similar to the CAPM, but assumptions are not so restrictive
- Assumptions:
- Uses multiple risk factors, and can use whichever you want (Oil price factor, interest rate, market portfolio factor, etc.)
- Well diversified portfolios can be created to eliminate non-systematic risk.
- Basically, APT combines a bunch of well diversified portfolios (called a “Factor Portfolio”) and uses all of them to form an efficient portfolio.
- \(E[R_i] = r_f + \sum \beta_i^{F_n}(E[R_{F_n}]-r_f)\)
- \(E[R_i]\) = Expected return of the asset portfolio
- \(\beta_i^{F_n}\) = Factor betas of asset i, measures sensitivity of the asset to a particular factor.
- \(R_{F_n}\) = The return on factor portfolio n
- If they say that hr portfolio is “Self Financing” we rewrite this as:
- \(E[R_i] = r_f + \sum \beta_i^{F_n} E[R_{F_n}]\)
- This means we are shorting the risk free asset to fund the asset portfolio
- Fama-French-Carhart (FFC)
- The most commonly used multi-factor model. Research shows it’s better than the CAPM
- Considers 4 factors: Market, Market Cap, Book-to-Market Ratios, and Momentum
- Each of these 4 factors are considered “Self-Financing” (meaning we borrow from something else and invest in the other)
- \(E[R_i]=r_f + \beta_i^{Mkt}(E[R_{mkt}]-r_f) + \beta_i^{SMB}E[R_{SMB}] + \beta_i^{HML}E[R_{HML}] + \beta_i^{PR1YR}E[R_{PR1YR}]\)
- Mkt = Market portfolio
- SMB = Small-minus-big portfolio
- HML = High-minus-low portfolio
- PR1YR = Prior 1-year momentum portfolio
- Self Financing Mkt Portfolio
- Expected return = \(E[R_{Mkt}] - r_f\)
- Same as the CAPM model. Accounts for Equity risk. Short the rfr and long the mkt portfolio.
- Small-minus-big (SMB) Portfolio
- \(E[R_{SMB}] = E[R_{small}] - E[R_{big}]\)
- Buys small firms, and finances by shorting big firms. Historically this has been positive b/c small cap tends to outperform large cap.
- High-minus-low (HML) Portfolio
- \(E[R_{HML}] = E[R_{HBM}] - E[R_{LBM}]\)
- Buys high book-to-market ratio stocks and shorts low book-to-market stocks
- Book value = accounting value of the firm’s balance sheet, how much would be left in assets if the firm went out of business today
- Market value = price investors are willing to pay in the market for a share of stock. It reflects supply and demand
- Book-to-Market Ratio = (Book value of firm) / (Mkt Value of firm)
- If > 1: then the actual worth exceeds the mkt value (Undervalued) (means it’s a value stock)
- If < 1: then the actual worth is below the mkt value (Overvalued) (means it’s a growth stock)
- Prior 1-year Momentum (PR1YR) Portfolio
- \(E[R_{PR1YR}] = E[R_W] - E[R_L]\)
- \(R_W\) = Last year’s winners
- \(R_L\) = Last year’s losers
- This factor accounts for momentum. It buys the top 30% stocks, and shorts the bottom 30% stocks.
- \(E[R_{PR1YR}] = E[R_W] - E[R_L]\)
- Fama-French (No Carhart)
- Same as the FCC, but we drop the momentum factor
- \(E[R_i]=r_f + \beta_i^{Mkt}(E[R_{mkt}]-r_f) + \beta_i^{SMB}E[R_{SMB}] + \beta_i^{HML}E[R_{HML}]\)
Efficient Market Hypothesis
| Form | Past Market Data | Public Information | Private Information |
|---|---|---|---|
| Weak | \(\checkmark\) | ||
| Semi-strong | \(\checkmark\) | \(\checkmark\) | |
| Strong | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) |
- Past Market Data = Current security prices already reflect all historical stock prices
- Public Information = Current security prices already reflect info from press
Private Information = Current security prices already reflect info from inside private information and in-depth security analysis
- EMH says you can’t beat the market by knowing the above things because current mkt prices already reflect that information. But people vary in how much they believe it.
For the exam, you may need to read a study and determine which form of the EMH the study supports or negates.
- Reasons the Market Portfolio is Not Efficient:
- Not all investors have homogeneous expectations (diff risk preferences and information)
- Investors are irrational (and misinterpret info)
- Investors carry portfolios that are mean-variance inefficient (don’t invest on the efficient frontier)
- Proxy Error = The true market portfolio is every single asset tradable, it’s hard to find a fund like this so we use a proxy like the S&P 500, and there is some error.
- Other behavioral biases
- Non-tradable wealth = If you work in a certain industry, and your industry goes down, your human capital in that industry is non-tradable
Investment Risk
Risk measures
Variance and Semi-Variance
Variance = Average squared deviation around the mean
- Semi-Variance
- Investers may only care about downside risk
- Semi-variance = Average squared deviation below the mean
- Semi Variance = \(\sigma^2_{SV} = E[min(0, R - E[R])^2] = \frac{1}{n}\sum min(0, R_i - E[R])^2\)
- For the E[R], still use all the + and - values, but then in the variance formula use 0 where \(R_i - E[R]>0\)
- Regular variance = Downside Semi-variance + Upside Semi-variance
Value at Risk (VaR)
- Simply the value (\(X\)) that gives you some percentile (\(\alpha\) or \(p\))
- \(VaR_\alpha(X)=(\pi_\alpha=X)=F^{-1}_X(\alpha)\)
- We usually just assume a normal distribution, so we would find a z-score, and plug it into the inverse normal CDF calculator
- If they say, what is the value at risk at the 95% percentile (it means what is the value X that is at the 95% percentile):
- So the value at risk at the 95% percentile is:
- \(VaR_.95(X) = \pi_.95 = N^{-1}(.95)=1.64485\)
- So we use that as our z-score and use the problem mean and variance to find the actual value where it’s the 95% percentile
- \(VaR_.95(X) = \pi_.95 = N^{-1}(.95)=1.64485\)
- So the value at risk at the 95% percentile is:
- If they say, the value at risk at the 95% percentile for X is 100. This means:
- \(VaR_{.95}(X) = 100\)
- => P(X < 100) = .95
Tail Value at Risk
- AKA: Conditional Tail Expectation (CTE), Expected Shortfall
- The expected value, given that a loss is above some percentile
- Formula is written slightly different if dealing with an expected loss or gain amount
- When X represents the expected LOSS Amount
- \(TVaR_\alpha(X) = E[X|X>\pi_\alpha] = \frac{\int_{\pi_\alpha}^\infty x f(x) dx}{P(X>\pi_\alpha)} = \frac{\int_{\pi_\alpha}^\infty x f(x) dx}{1-\alpha}\)
- We are concerned about the adverse scenario at the HIGH end (right tail)
- When X represents the expected GAIN amount:
- \(TVaR_\alpha(X) = E[X|X \leq \pi_\alpha] = \frac{\int^{\pi_\alpha}_{-\infty} x f(x) dx}{P(X \leq \pi_\alpha)} = \frac{\int^{\pi_\alpha}_{-\infty} x f(x) dx}{\alpha}\)
- We are concered about the adverse scenario at the LOW end (left tail)
- TVaR \(\leq\) VaR
Coherent Risk Measures
- A Risk Measure is “Coherent” if the following 4 properties are present:
- (Let X and Y be 2 RV’s w/ risk measures g(X) and g(Y))
- Translation Invariance: \(g(X+c)=g(X)+c\)
- Adding a positive amount of risk adds an equivalent amount of risk to the risk measure
- Positive Homogeneity: \(g(cX) = cg(X)\)
- Multiplying an amount of rrisk, increases the risk measure in a proportional manner
- Subadditivity: \(g(X+Y)=g(X) + g(Y)\)
- Diversification benefits from combining risks as long as they aren’t perfectly correlated
- Monotonicity: If X < Y then \(g(X) < g(Y)\)
- If one risk exceeds the other, then so will the corresponding risk measure
- How risk measures measure up
- Variance and Semi-Variance do not satisfy any of these conditions
- VaR typically satisfies all conditions except subadditivity. But if the underlying distribution is normal, then VaR satisfies them all
- TVaR is Coherent b/c it satisfies all conditions
Project Risk analysis
- We often use NPV to determine on investing in a project (\(NPV = \sum \frac{CF_i}{(1+i)^t}\))
- This has 2 moving levers (Future Cash Flows, and Cost of Capital (interest rate))
- 4 methods to estimating these 2 moving levers:
- Break even analysis
- A project breaks even of the NPV = 0
- IRR = the rate of return that makes the NPV = 0
- Sensitivity Analysis
- You have a worst case, medium case, and best case scenarios. You change 1 variable at a time (units sold, price, tax rate, cost of capital, etc.), and see how changing each one effects the NPV.
- Scenario Analysis
- You have a worst case, medium case, and best case scenarios. But this time you adjust all variables in each category, find the probability of each scenario, and calculate the E[NPV].
- Monte Carlo Simulations
- Define a probability distribution for each input, and then run simulations on each input distribution and calculate thousands of NPV’s to find the average
Real Options
- When you have the right, but not the obligation, to make a particular business decision in the future after new info becomes available. This creates an option in real life.
- Will need to draw decision trees and find the payoffs, probabilities, and NPV.
- Timing Option
- When you have the option to delay an investment, with some consequences but you also get new information
- Main formula:
- Value of real option = NPV(with option) - NPV(without option)
Behavioral Biases
Pre-Post money valuation
- Pre-Money Valuation = value of a firm before a funding round
- Post-Money Valuation = value of a firm after a funding round
- Post-Money Valuation = (Pre-Money Valuation) + (Amount invested)
- Post-Money Valuation = (Number of shares After the funding round)*(Pre-Money price per share)
- Percentage Ownership: 3 Formulas:
- = \(\frac{Amount~Invested}{Post-Money~Valuation}\)
- = \(\frac{(\#~Shares~Owned)*(Pre-Money~Price~per~Share)}{Post-Money~Valuation}\)
- = \(\frac{\#~of~Shared~Owned}{Total~\#~of~Shares}\)
Capital Structure
- A lot of reading to do in this section. Best to reference the text (Berk and DeMarzo). Or look at last few pages of CA study guide.
Some random things:
Corporate Debt (Public Debt)
- Corporate Bonds:
- Notes: Unsecured, shorter maturities
- Debentures: Unsecured
- Subordinated Debenture: debt w/ lower seniority than existing debt
- Mortgage & Asset-backed Bonds: Secured, greater seniority in case of default
- Most corporate bonds pay coupons semi-annually, but not always
- Bearer Bonds: owner of the bond must physically hold the certificate to receive payment. This is like having cash.
- Registered Bonds: Issuer of the bond has a list of all holders, there is no physical bond certificate. This is how bonds are today.
- Corporate Bonds:
Some things on Systematic and Non systematic Risk:
- The total risk in a portfolio can be broken down into two components:
- Systematic risk. Also known as common, market, or nondiversifiable risk. It is a risk inherent in an entire market.
- Nonsystematic risk. Also known as firm-specific, independent, idiosyncratic, unique, or diversifiable risk. It is a risk inherent in a specific company or industry.
- Diversification reduces a portfolio’s total risk by averaging out nonsystematic fluctuations. Systematic risk cannot be avoided through diversification.
- Two important principles:
- Because investors can eliminate nonsystematic risk “for free” by diversifying their portfolios, they do not require a risk premium for holding nonsystematic risk. The risk premium for nonsystematic risk is zero.
- Because investors cannot eliminate systematic risk “for free” by diversifying their portfolios, they demand a risk premium for holding systematic risk. As a result, the risk premium of a security is determined by its systematic risk rather than its total risk.
- As a consequence of the previous point, under the two asset pricing models covered in this course (CAPM, multi-factor model), only systematic risk matters; nonsystematic risk is irrelevant. This is because investors can eliminate nonsystematic risk “for free” by diversifying their portfolios, so they are not compensated for holding nonsystematic risk.
- Beta is a measure of systematic risk.
- The total risk in a portfolio can be broken down into two components: